Remarks on the balanced metric on Hartogs triangles with integral exponent

In this paper we study the balanced metrics on some Hartogs triangles of exponent γ ∈ ℤ+, i.e. equipped with a natural Kähler form with where μ = (μ1, …, μn), μi > 0, depending on n parameters. The purpose of this paper is threefold. First, we compute the explicit expression for the weighted Bergman kernel function for (Ωn(γ),g(μ)) and we prove that g(μ) is balanced if and only if μ1 > 1 and γμ1 is an integer, μi are integers such that μi ≽ 2 for all i = 2, …, n − 1, and μn > 1. Second, we prove that g(μ) is Kähler-Einstein if and only if μ1 = μ2 = … = μn = 2λ, where λ is a nonzero constant. Finally, we show that if g(μ) is balanced then (Ωn(γ),g(μ)) admits a Berezin-Engliš quantization.